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Math 166Z Homework # 11 Due Thursday April 28th 1. Approximate each function below with the nth order Taylor polynomial Pn (x) centered at the number a. Then estimate the accuracy of your approximation for x in the given interval. π (a) f (x) = sin x, a = π4 , n = 5, 0 ≤ x ≤ 2 (b) f (x) = ln x, a = 4, n = 3, 3 ≤ x ≤ 5 2. For each set of parametric equations below: (1) Sketch the curve (2) Eliminate the parameter to find the Cartesian equation of the curve dy d2 y (2) Find and dx dx √ (a) x = t, y = 1 − t (b) x = sin θ, y = cos θ, 0 ≤ θ ≤ π (c) x = 3t2 , y = 2 + 5t, 0 ≤ t ≤ 2 3. Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. π (a) r = cos θ + sin θ, θ = 4 (b) r = ln θ, θ = e 4. Find the area of the region that is bounded by the given curve and lies in the given sector. π (a) r = 2 cos θ, 0 ≤ θ ≤ 2 π θ (b) r = e , 0 ≤ θ ≤ 6 1 5π (c) r = , 0 ≤ θ ≤ θ 6 5. Find the area of the region enclosed by one loop of the curve. (a) r = sin 3θ (b) r = 2 cos 4θ